Question

What are the extrema of `f(x)=x/(x-2)` on the interval [-5,5]?

Answer 1

There are no absolute extrema, and the existence of relative extrema depends on your definition of relative extrema.

Explanation:

`f(x) = x/(x-2)` increases without bound as `xrarr2` from the right.
That is: `lim_(xrarr2^+)f(x)=oo`
So, the function has no absolute maximum on `[-5,5]`

`f` decreases without bound as `xrarr2` from the left, so there is no absolute minimum on `[-5,5]`.

Now, `f'(x) = (-2)/(x-2)^2` is always negative, so, taking the domain to be `[-5,2) uu (2,5]` , the function decreases on `[-5,2)` and on `(2,5]`.

This tells us that `f(-5)` is the greatest value of `f` nearby considering only `x` values in the domain. It is a one-sided relative maximum. Not all treatments of calculus allow one sided relative extrema.

Similarly, if your approach allow one-sided relative extrema, then #f(5) is a relative mimimum.

To help visualize, here is a graph. The restricted domain graph is solid and the endpoints are marked.

The natural domain graph extends into the dashed line part of the picture.